348 Mr. Graves's Explanation of a remarkable Paradox 



tive or negative, cannot hold good for z of an opposite sign, 



unless c = — . 

 c 



Inner Temple, July 1836. 



P.S. It is not out of place to mention, that I am gratified 

 with the view which Professor De Morgan has taken in the 

 last Number of this Magazine (October 1836, vol. ix. p. 252,) 

 of my researches on logarithms, and that I agree with him in 

 considering my results rather upon the whole as extensions 

 or (as I should say) completions, than as corrections of what 

 had before been accomplished. He has also properly noticed 

 an oversight I committed in not observing the distinction he 

 drew in his Calculus of Functions, with reference to the possi- 

 bility of obtaining the most general solution, between func- 

 tional equations where there are and where there are not in- 

 dependent variables. I may be permitted, however, to assent 

 to his remarks on some other points with some qualifications, 

 which may seem over-nice and pedantic, but are required by 

 the delicacy of the subject, and I wish to prefix some expla- 

 nation relative to the actual progress or improvement which 

 I consider this branch of science to have received from my 

 researches. The deficiencies in the ordinary theory which I 

 have endeavoured to supply are the following : first, I found no 

 formula which assigned even one value of a*, much less all of 

 them, when a and x were imaginary ; secondly, I considered 

 that as —2 was a value of 44, \ would be admitted to be, or 

 at least deserved to be reckoned a logarithm of —2 to the 

 base 4, and yet in no formula which 1 had met with for the 

 4-logs of —2 was any such quantity as \ included; thirdly, 

 I observed generally great laxity, not to say inaccuracy, in 

 the use of ambiguous exponential expressions, and saw 

 equations employed without apparent restriction where, 

 perhaps, the two sides had but one value in common. 

 For instance, the equation gi + V-i 2wi7r— g ;§ not correct 

 without restricting the meaning of the left-hand side, for 

 though every quantity included in the formula \+ V —\2 irn: 

 is an ^--log of ^, e^ +-v/— i 'i-m-K has an infinite number of real 

 values besides e for any given ot, except m = 0. Hence, I 

 scruple to call e i + V— i2»i tt merely a general algebraic form 

 of e, and think it necessary to devise a notation to characterize 



that particular value of e^+'v^-^^"*'^ which is equal to e. 



1+ V— i Im-r 



The still more general form ^l+\/-i2n» ^^^^ equally 



